This paper establishes a structure-preserving numerical scheme for the Cahn--Hilliard equation with degenerate mobility. First, by applying a finite volume method with upwind numerical fluxes to the degenerate Cahn--Hilliard equation rewritten by the scalar auxiliary variable (SAV) approach, we creatively obtain an unconditionally bound-preserving, energy-stable and fully-discrete scheme, which, for the first time, addresses the boundedness of the classical SAV approach under $H^{-1}$-gradient flow. Then, a dimensional-splitting technique is introduced in high-dimensional cases, which greatly reduces the computational complexity while preserves original structural properties. Numerical experiments are presented to verify the bound-preserving and energy-stable properties of the proposed scheme. Finally, by applying the proposed structure-preserving scheme, we numerically demonstrate that surface diffusion can be approximated by the Cahn--Hilliard equation with degenerate mobility and Flory--Huggins potential when the absolute temperature is sufficiently low, which agrees well with the theoretical result by using formal asymptotic analysis.wn theoretically by formal matched asymptotics.
翻译:本文为可变性下降的Cahn- Hilliard 方程式建立了一个结构保存数字方案。 首先,我们创造性地获得了一个无条件的约束性、能源稳定性和完全分解性方案,这是首次在 $H<unk> -1} 美元 梯度流下处理经典 SAV 方程式的界限性。 然后,在高维情况下引入了方位分离技术,大大降低了计算复杂性,同时保留了原始结构特性。我们介绍了数值实验,以核实拟议方程式的约束性、节能和可能量特性。最后,通过应用拟议的结构保留方案,我们从数字上表明,当绝对温度足够低时,可与具有低流动性和软性- 粗度潜力的Cahn- Hilliard 方程式相近,这与理论结果是一致的,同时使用正式的模拟模拟分析。</s>